An explosion in the number of wireless broadband users has led to a severe spectrum shortage in the conventional cellular bands. The demand for cellular data services is expected to grow at a staggering rate, necessitating orders of magnitude increases in wireless capacity [see, e.g., Refs. 1, 2]. Millimeter wave (mmW) frequencies at 28, 38, and 60 GHz have been attracting growing attention as a possible candidate for next-generation microcellular networks [see, e.g., Refs. 3-5]. This band offers orders of magnitude greater spectrum and also allows for building high dimensional antenna arrays for further gains via beamforming and spatial multiplexing. Devices based on mmW have already hit the market, but are limited by their use of highly directional horn antennas to enable short-range, line-of-sight links, within a controlled and static environment, such as in a data center [see, e.g., Ref. 6]. Since such an environment and conditions are very difficult—if not impossible—to achieve in a practical system implement, there is a need for building mmW orthogonal frequency division multiplexing (OFDM) systems “in the wild,” e.g., where line-of-sight is not always available, SNRs are lower, mobility is enabled, and the use of static directional antennas is infeasible.
One of the biggest challenges [see, e.g., Ref. 7] in building such mmW OFDM systems is performing accurate channel estimation and equalization because the channel varies rapidly across OFDM subcarriers and/or symbol indices (see FIG. 1). As discussed in more detail herein, there are many reasons for these variations, such as multipath interference, Doppler shifts, and time frequency offsets, which are further exacerbated in mmW frequencies, leading to very rapid channel variations. Existing interpolation mechanisms for interpolating between channel estimates at widely-spaced pilot symbols (9) are not capable of tracking these variations within reasonable bounds of error without requiring prohibitively high pilot densities. The resulting self-noise becomes very significant, and it cannot be overcome by traditional methods of increasing the SNR such as beamforming or increasing the transmit/receive gains. This is a missing piece of the puzzle that various embodiments described herein aim to provide.
There are various causes of channel variations, and these may be exacerbated in mmW. Such channel variations may lead to interpolation errors in the real and imaginary components of the channel. In the case of tightly-spaced modulation constellations such as 64-QAM, even a 10% error in the channel estimate can lead to symbols being incorrectly decoded.
Doppler Shifts:
When there is relative motion between the transmitter and receiver, the frequency of the signal as perceived by the receiver is shifted from the true frequency, leading to apparent channel phase rotations across symbol indices. Since the amount of this Doppler shift is linear in the carrier frequency [see, e.g., Ref. 10], the rate of change of the channel can be an order of magnitude faster in the mmW range (e.g., >28 GHz) relative to conventional cellular or WiFi systems having carrier frequencies below 3 GHz. FIG. 2(a) illustrates the phase for an exemplary line-of-sight wireless channel at (2.4, 60) GHz carrier frequency with Doppler shift caused by a receiver speed of 30 mph. It is apparent from FIG. 2(a) that higher carrier frequencies lead to faster varying channels for moving receivers and/or transmitters.
Bursty Directional Transmissions:
Due to their small wavelength, the omni-directional path loss of mmW signals can be easily 20 to 30 dB higher than conventional cellular transmissions (a consequence of Friis Law [see, e.g., Ref. 10]). As a result, long-range transmissions depend on highly directional beamforming to different users. A consequence of directional transmissions is that, unlike traditional cellular systems where pilot or reference signals can be continuously tracked in both the uplink and downlink, mmW transmissions to any mobile unit will be much more bursty and intermittent. As a result, receivers must re-synchronize each MAC frame, leading to significant residual timing and carrier frequency offsets, as elaborated below:
Timing Offsets:
These offsets can be limited to about 3% of the symbol duration [see, e.g., Ref. 11], but are still sufficient to cause phase rotations in the channel across subcarrier indices. FIG. 2(b) illustrates an exemplary channel that would result from timing offsets of 3 and 7 samples, corresponding to 1.17% and 2.73% of a 256-sample symbol. FIG. 2(b) illustrates the general principle that larger timing offsets lead to more rapid variations in the channel phase across subcarriers.
Residual Frequency Offsets (RFO):
These offsets, which are due to small offsets between the nominal and actual carrier frequencies generated in the transmitter and/or receiver, can be limited to about 1% of the subcarrier bandwidth [12] but still may cause significant variations in the channel across symbol indices. FIG. 2(c) illustrates an exemplary channel response when the channel bandwidth is 20 MHz, the OFDM FFT size is 256, the subcarrier bandwidth is 78.125 kHz, and the RFO is (0.5, 1.0)% of the subcarrier bandwidth. Larger values of RFO lead to more rapid channel variations across symbol indices.
Multipath Estimation on Wide Bandwidths:
One of the main benefits of mmW systems is the possibility of using wide bandwidths. However, use of any spectrum demands an accurate channel estimate across the entire frequency band. The rate of variation of the channel across frequency is proportional to the delay spread, which is the difference in time delay of arrivals between different paths. Measurements described in Ref. [1] have demonstrated that typical delay spreads in mmW range from 200 ns to as high as 1 μs. FIG. 2(d) illustrates the response of a simple two-ray multipath channel with a delay spread of 200 ns, a power difference between rays of 10 dB, channel bandwidths of 20 MHz and 100 MHz, OFDM FFT size of 256, and a cyclic prefix (CP) of 64 samples to eliminate inter-symbol interference (ISI). The exemplary results shown in FIG. 2(d) illustrate that larger channel bandwidths lead to more rapid variations across subcarrier indices. The presence of strongly directional multipath components common in mmW systems (see, e.g., Ref. 13) makes these variations even more rapid.
In summary, the mmW channel can change rapidly across OFDM subcarrier indices due to multipath interference and timing offsets as well as across symbol indices due to RFO and Doppler). These effects are exacerbated in mmW due to the rich multipath diversity, high carrier frequencies, and large channel bandwidths. Interpolation errors increase when the underlying channel exhibits more rapid variances, leading to lower quality channel estimates at the receiver.
As mentioned above, existing channel interpolation techniques are generally inadequate to track and compensate for the channel variations described hereinabove. Five of these existing techniques are briefly described below.
Parametric Estimation:
FIGS. 2(a), 2(b), and 2(c) illustrate exemplary channel variations caused by Doppler shifts and timing/frequency offsets that are, e.g., perfectly sinusoidal. As such, the channel can be reconstructed by estimation of two parameters of a sinusoid: a) amplitude; and b) initial phase. However, sinusoidal reconstruction is not robust in the presence of multipath interference prevalent in mmW multipath channels, which causes non-sinusoidal variations illustrated in the exemplary FIG. 2(d).
Linear Interpolation:
This exemplary technique is robust, as it does not make any prior assumptions about the nature of channel variations (unlike sinusoidal reconstruction). The channel estimate at a data or non-pilot resource element (RE) is derived from the linear interpolant, which is the straight line that connects the channel estimates at the nearest two pilot locations. The real and imaginary components of the channel h are interpolated independently. While linear interpolation is popular for its computational efficiency, its main weakness is large interpolation errors when the interpolant between the two pilot locations spans over a crest or trough in the underlying channel component.
Polynomial Interpolation:
The channel can be reconstructed by estimating the parameters of an nth order polynomial that fits the channel estimates derived from the pilots. Since n is proportional to the OFDM FFT size and the packet size, it can be very large, rendering this technique computationally infeasible for practical mmW systems.
2D Triangulation:
Widely implemented for its computational efficiency and simplicity, 2D triangulation simply calculates the weighted average of the channel estimates derived from the three nearest pilot locations. The real and imaginary components of the channel are independently interpolated. The main weakness of this scheme is large interpolation errors when: a) the pilots used to estimate the channel at an RE span a crest or trough (in 2D space); and b) the channel exhibits variations at different rates across subcarrier and symbol indices.
Decision Directed Feedback Equalization:
In DDFE, the receiver maintains hi, the estimate of the channel on each subcarrier i. The initial values of each hi are calculated from the packet preamble. When a symbol ri is received, it is estimated as
            s      i        =                  r        i                    h        i              ,and matched to the nearest symbol in the constellation (ŝi). The channel estimate is then updated as
            h      i        =                  r        i                    s        i              ,and is used for the next symbol. DDFE works well when channel variations across time are slow, but fails when there is considerable RFO or Doppler. The advantage of DDFE is that if the channel exhibits rapid changes across subcarriers and is time invariant, the channel estimate on every subcarrier is perfect, since the estimates are derived from the preamble rather than being interpolated from the pilots.
Whittaker-Shannon Sinc Interpolation:
According to theory [see, e.g., Refs. 15 and 16], Sinc interpolation is an optimal method to construct a continuous bandlimited function from a set of sampled known real values of that function; the real and imaginary components are therefore interpolated independently. In a 2D grid of subcarriers and symbol indices, the pilot REs are replaced by impulses, scaled by the value of the channel at that RE. The resulting grid of REs is passed through an ideal low-pass filter (which is the equivalent of convolution with a Sinc function) to arrive at the channel estimates at each data (non-pilot) RE. While Sinc interpolation is optimal in a bandlimited sense, the underlying channel variations are not always bandlimited, leading to interpolation errors. Another weakness of Sinc interpolation is the Gibbs phenomenon that leads to large interpolation errors at the edges of the packet (FIG. 1). There are techniques to mitigate the Gibbs phenomenon (such as the Fejér summation [see, e.g., Ref. 17] or wavelet transform [see, e.g., Ref. 18]), but their study is out of the scope of the description provided herein.
Thus, there may be a need to address at least some of the inadequacies, issues, and/or concerns with existing OFDM channel estimate interpolation techniques described herein.